Can you Weight Paint differently on Linked Duplicates?
Select your object, go in Edit mode and assign a vertex group to the faces that are supposed to emit the particles. Then in the Properties panel > Particles, create your particle settings and in the Vertex Groups submenu > Density field, choose the vertex group you've created.

Link-duplicate your object (AltD), go in Edit mode and assign a new vertex group to the faces that are supposed to emit the particles on this second object. Then in the Vertex Groups submenu > Density field, choose this new group.

The objects are still linked but the particles emission will happen on 2 different parts.